Complete negatively curved immersed ends in $\Bbb R^3$

TL;DR

This paper extends Efimov's Theorem to show that complete immersed surfaces in three-dimensional space with strongly negatively curved ends cannot exist if their total absolute curvature is infinite.

Contribution

It sharpens Efimov's Theorem by proving non-existence of certain complete immersed surfaces with negatively curved ends and infinite total curvature in D space.

Findings

No complete immersed surface with infinite total absolute curvature and negative Gaussian curvature bounded away from zero exists.

The result applies to complete Hadamard immersed surfaces with curvature bounded away from zero outside a compact set.

Extends classical non-existence results for negatively curved surfaces in D.

Abstract

This paper extends, in a sharp way, the famous Efimov's Theorem to immersed ends in $\real^3$. More precisely, let $M$ be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of $M$ into $\Bbb R^3$ satisfying that $\int_M |K|=+\infty$ and $K\le-\kappa<0$, where $\kappa$ is a positive constant and $K$ is the Gaussian curvature of $M$. In particular Efimov's Theorem holds for complete Hadamard immersed surfaces, whose Gaussian curvature $K$ is bounded away from zero outside a compact set.